The remainder of what comes is interpreting the word “like” in the initial question above, as it can be done in a variety of settings. We can say, for instance, that groups with expanders in them do not coarsely embed into a Hilbert space (one of the initial motivations for constructing them). However, such groups also give us the only known counterexamples for the Baum–Connes conjecture with coefficients - this is present in the paper of Higson–Lafforgue–Skandalis [1]} with very general statements that rely on a very basic construction they exploit over and over, in particular in the case of groups that contain expanders.
